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arXiv:2512.08687

Non-Hermitian symmetry breaking and Lee-Yang theory for quantum XYZ and clock models

Tian-Yi Gu, Gaoyong Sun

quant-phSubmitted Dec 9, 2025• Announced Dec 10, 2025• 0 citations
v1 · Dec 9, 2025v2 · Feb 13, 2026v3 · Aug 28, 2026
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Abstract

Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework is extended to characterize quantum phase transitions in arXiv:2509.20258 by introducing the concepts of non-Hermitian symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY model, the XXZ model, the XYZ model, and the $\mathbb{Z}_3$ clock model in one dimension, subject to complex external magnetic field. For the XY, XXZ and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the $\mathbb{Z}_3$ clock model, the complex field splits the real part of the ground-state energy between the neutral sector ($q=0$) and the charged sectors ($q=1,2$), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors, and the finite-size scaling of these zeros produces critical exponents fully consistent with analytical predictions.

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7 pages, 4 figures

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Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework is extended to characterize quantum phase transitions in arXiv:2509.20258 by introducing the concepts of non-Hermitian symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY model, the XXZ model, the XYZ model, and the Z3\mathbb{Z}_3Z3​ clock model in one dimension, subject to complex external magnetic field. For the XY, XXZ and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the Z3\mathbb{Z}_3Z3​ clock model, the complex field splits the real part of the ground-state energy between the neutral sector (q=0q=0q=0) and the charged sectors (q=1,2q=1,2q=1,2), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors, and the finite-size scaling of these zeros produces critical exponents fully consistent with analytical predictions.
Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework was extended to characterize quantum phase transitions of quantum Ising models by introducing the concepts of non-Hermitian parity-symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY, the XXZ, the XYZ, and the Z_{p} clock models in one dimension, subject to a complex magnetic field. For the XY, XXZ, and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the Z_{3} clock model, the complex field splits the real part of the ground-state energy between the neutral sector (q=0) and the charged sectors (q=1,2), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors. For the Z_{4} and Z_{5} clock models, the ground states are instead projected to oscillate between the neutral sector (q=0) and the charged sectors q=2 and q=1, respectively, giving rise to fidelity zeros. Finite-size scaling of these zeros yields critical exponents in full agreement with analytical predictions, demonstrating that this approach is applicable not only to the Ising model with Z_{2} symmetry, but also to more general Heisenberg-type models and systems with higher discrete symmetries.
Lee-Yang theory is central to the analysis of thermal phase transitions. However, the underlying mechanism of the theory and the nature of Lee-Yang zeros in quantum many-body systems remains elusive. Here, we develop a unified framework for understanding quantum phase transitions from fidelity zeros induced by symmetry breaking. These zeros, arising from transitions between symmetry sectors, obey the Lee-Yang theorem and give rise to fidelity edges near critical points. Quantum criticality is further characterized through the finite-size scaling of fidelity zeros. As concrete examples, we analytically and numerically investigate fidelity zeros in one- and two-dimensional ferromagnetic Ising models under a complex magnetic field. Our results provide new insights into the mechanism of Lee-Yang theory and open avenues for exploring unexplored landscapes of phase transitions in quantum many-body systems.
Lee–Yang theory clearly demonstrates where the phase transition of many-body systems occurs and the asymptotic behavior near the phase transition using the partition function under complex parameters. The complex parameters make the direct investigation of Lee–Yang theory in practical systems challenging. Here we construct a non-Hermitian quantum system that can correspond to the one-dimensional Ising model with imaginary parameters through the equality of partition functions. By adjusting the non-Hermitian parameter, we successfully obtain the partition function under different imaginary magnetic fields and observe the Lee–Yang zeros. We also observe the critical behavior of free energy in vicinity of Lee–Yang zero that is consistent with theoretical prediction. Our work provides a protocol to study Lee–Yang zeros of the one-dimensional Ising model using a single-qubit non-Hermitian system.
In this work, we investigate many-body phase transitions in a one-dimensional anisotropic XY model subject to a complex-valued transverse field. Within the biorthogonal framework, we calculate the ground-state correlation functions and entanglement entropy, confirming that their scaling behavior remains identical to that in the Hermitian XY model. The preservation of Hermitian phase transition features in the non-Hermitian setting is rooted in the persistence and emergence of symmetries and their breaking. Specifically, the ferromagnetic (FM) phase arises from the breaking of a Z2Z_2Z2​ symmetry, while the Luttinger liquid (LL) phase is enabled by the emergence of a U(1)U(1)U(1) symmetry together with the degeneracy of the real part of the energy spectrum. The nontrivial topology of the LL phase are characterized by the winding number around the exceptional point (EP). Given that non-Hermitian systems are inherently open, this research opens a new avenue for exploring conventional quantum phase transitions that are typically vulnerable to decoherence and environmental disruption in open quantum systems.
We present a comprehensive theoretical framework for quantum criticality in the non-Hermitian detuned PXP model, and establish the complete phase diagram, which had remained elusive in previous studies. Starting from a numerically identified phase transition point, we construct an exact second-order phase transition boundary through a similarity transformation in the real-energy regime. By introducing the biorthogonal entanglement entropy and biorthogonal Loschmidt echo, we demonstrate from both equilibrium and nonequilibrium perspectives that this transition belongs to the Ising universality class. Using the correlation function, we further distinguish between confined and deconfined phases within the PT\mathcal{PT}PT-symmetric region. In the complex-energy regime, we identify both a full PT\mathcal{PT}PT transition and a first-excited-state PT\mathcal{PT}PT transition, respectively. Moreover, we identify the location of the Yang-Lee edge singularity (YLES) using both the associated-biorthogonal and self-normal Loschmidt echoes, and extract the corresponding critical exponent, which agrees with the predictions of nonunitary conformal field theory. Finally, we propose an experimental scheme to observe the YLES in Rydberg atomic arrays, which offers a promising route to exploring non-Hermitian critical phenomena and singularities in future experimental settings.
Determining the phase diagram of interacting quantum many-body systems is an important task for a wide range of problems such as the understanding and design of quantum materials. For classical equilibrium systems, the Lee-Yang formalism provides a rigorous foundation of phase transitions, and these ideas have also been extended to the quantum realm. Here, we develop a Lee-Yang theory of quantum phase transitions that can include thermal fluctuations caused by a finite temperature, and it thereby provides a link between the classical Lee-Yang formalism and recent theories of phase transitions at zero temperature. Our methodology exploits analytic properties of the moment generating function of the order parameter in systems of finite size, and it can be implemented in combination with tensor-network calculations. Specifically, the onset of a symmetry-broken phase is signaled by the zeros of the moment generating function approaching the origin in the complex plane of a counting field that couples to the order parameter. Moreover, the zeros can be obtained by measuring or calculating the high cumulants of the order parameter. We determine the phase diagram of the two-dimensional quantum Ising model and thereby demonstrate the potential of our method to predict the critical behavior of two-dimensional quantum systems at finite temperatures.